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Comment on "Numerical estimates of the spectrum for anharmonic PT symmetric potentials"

Amore, Paulo, Francisco Marcelo Fernández

Published Mar 1, 2013• 3 citations
OpenAlex

Abstract

We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947)

Related papers

Comment on ‘Numerical estimates of the spectrum for anharmonic PT symmetric potentials’

Paolo Amore, Francisco Marcelo Fernández

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947).

arXiv:1209.6357•Physica Scripta•Mar 8, 2013•1 citation

Comment on: ‘PT-/non-PT-symmetric and non-Hermitian Hellmann potential: approximate bound and scattering states with anyℓ-values’

Paolo Amore, Francisco Marcelo Fernández

We carry out an analysis of the paper published in Arda and Sever (2014 Phys. Scr. 89 105204 ) and reveal several unclear points. In the first place, the derivation of the main equations does not appear to be correct. In the second place, the expression for the energy does not appear to yield the results reported by the authors. In the third place, they failed to indicate the references reporting the eigenvalues chosen for comparison. In the fourth place the eigenvalues reported by the authors do not show the same order as those of the Hellmann potential thus leading to a different underlying physics. What is more: the spectrum of the modified model is qualitatively different from the one supported by the Hellmann potential.

Physica Scripta•Jun 16, 2015•0 citations

Comment on : Scarcity of real discrete eigenvalues in non-analytic complex PT -symmetric potentials.Z.Ahmed ,Pramana J.Phys 73(2),323(2009)

Biswanath Rath

We notice contradictory statements of Z.ahmed on $\mathcal{PT}$ symmetry potential $V=\frac{x^{2}}{4}+igx|x|$ (g=2) The curve (dash-dot) shows only one zero whose corresponding energy eigenvalue is 1.72.This graphical representation contradicts the discussion (presented below the graph )which claims number of real discrete states between $0\leg\le2.3$ are 3.as $g=2$ <$g=2.3$ ,one should get number of real discrete states 3 instead of 1.However present calculation matrix diagonalisation pethod reflects 3.secondly the statement on complex Harmonic Oscillator $V_{H}=x^{@}+ix$ (which entails complete real discrete spectrum) as a singular example is not acceptable as we notice many complex $|mathcal{PT}$ symmetric potentials can be constructed ,which would reflect infinitr real discrete energy eigenvalues.

•Dec 20, 2015•0 citations

Comment on the article by Bender et al, J. Phys. A 35, L467 (2002)

Carlos R. Handy

The recent Letter by Bender, Berry, and Mandilara (2002, BBM) presents some interesting symmetry arguments which enable one to transform non-hermitian, PT invariant, (complex) polynomial potential hamiltonians, into secular equation representations with real coefficients. This achievement is claimed to ``demistify'' why these systems can admit real and/or complex eigenenergies. Two approximations underly their arguments, and weaken the implied significance of their work.

arXiv:math-ph/0208021•Aug 16, 2002•0 citations

Comment on ``PT-Symmetric versus Hermitian Formulations of Quantum Mechanics [hep-th/0511229]''

Mostafazadeh, Ali

We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.

arXiv:hep-th/0603059•Mar 8, 2006•2 citations

A numerical and analytical investigation into non-Hermitian Hamiltonians

Gert Jermia Cornelus Wessels

In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.

•Mar 1, 2009•4 citations
We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947)
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-xxx plane (Rep. Prog. Phys. {\bf 70} (2007) 947).
We carry out an analysis of the paper published in Arda and Sever (2014 Phys. Scr. 89 105204 ) and reveal several unclear points. In the first place, the derivation of the main equations does not appear to be correct. In the second place, the expression for the energy does not appear to yield the results reported by the authors. In the third place, they failed to indicate the references reporting the eigenvalues chosen for comparison. In the fourth place the eigenvalues reported by the authors do not show the same order as those of the Hellmann potential thus leading to a different underlying physics. What is more: the spectrum of the modified model is qualitatively different from the one supported by the Hellmann potential.
We notice contradictory statements of Z.ahmed on PT\mathcal{PT}PT symmetry potential V=x24+igx∣x∣V=\frac{x^{2}}{4}+igx|x|V=4x2​+igx∣x∣ (g=2) The curve (dash-dot) shows only one zero whose corresponding energy eigenvalue is 1.72.This graphical representation contradicts the discussion (presented below the graph )which claims number of real discrete states between $0\leg\le2.3$ are 3.as g=2g=2g=2 <g=2.3g=2.3g=2.3 ,one should get number of real discrete states 3 instead of 1.However present calculation matrix diagonalisation pethod reflects 3.secondly the statement on complex Harmonic Oscillator VH=x@+ixV_{H}=x^{@}+ixVH​=x@+ix (which entails complete real discrete spectrum) as a singular example is not acceptable as we notice many complex ∣mathcalPT|mathcal{PT}∣mathcalPT symmetric potentials can be constructed ,which would reflect infinitr real discrete energy eigenvalues.
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.